Polynomial models arise in numerous applications in signal processing dealing with the estimation of continuous functions of either space or time. The requirement to interpolate between discrete noisy data points is of particular interest. In this report, the general-order polynomial time-varying coefficient-estimation problem is formulated and a structure is indicated for implementation of the coefficient-estimation process. Both the maximum likelihood (ML) and maximum a posteriori (MAP) estimation coefficient estimators are derived and the corresponding analyses of variance are obtained. The cases of measurement data consisting of samples of either the polynomial directly or the polynomial slope in the presence of additive noise are considered. Finally, expressions for the sensitivity to uncompensated measurement bias of the estimator standard deviations are obtained.